University of Toronto's Symplectic Geometry Seminar

Abstract: We define a `Higgs field' for a four-dimensional spin^c-manifold to be a smooth section of its positive half-spinor bundle, transverse to the zero section, and defined only up to a positive functional factor. This is intended to be a generalization of almost complex structures on real four-manifolds, each of which may in fact be treated as a Higgs field without zeros for a specific spin^c-structure. The notions of totally real or pseudoholomorphic immersions of real surfaces in an almost complex manifold of real dimension four have straighforward generalizations to the case of a spin^c-manifold with a Higgs field. Since a spin^c-structure with a Higgs field exists on every orientable four-manifold, immersions of surfaces that are pseudoholomorphic relative to a Higgs field might conceivably be used as probing tools to study the topology of a large class of manifolds in dimension four, in analogy with Taubes's work in the symplectic case. Our results consist, first, in showing that totally real immersions of closed oriented surfaces in four-dimensional spin^c-manifolds with Higgs fields have, basically, the same properties as in the almost-complex case, and, secondly, in providing a description of all pseudoholomorphic immersions of such surfaces in the four-sphere endowed with a "standard" Higgs field.